Abstract

A quantitative synthetic Schlieren imaging (SSI) method based on fast Fourier demodulation is presented. Instead of a random dot pattern (as usually employed in SSI), a 2D periodic pattern (such as a checkerboard) is used as a backdrop to the refractive object of interest. The range of validity and accuracy of this “Fast Checkerboard Demodulation” (FCD) method are assessed using both synthetic data and experimental recordings of patterns optically distorted by small waves on a water surface. It is found that the FCD method is at least as accurate as sophisticated, multi-stage, digital image correlation (DIC) or optical flow (OF) techniques used with random dot patterns, and it is significantly faster. Efficient, fully vectorized, implementations of both the FCD and DIC/OF schemes developed for this study are made available as open source Matlab scripts.

Notes

Acknowledgements

I am indebted to Antonin Eddi and Emmanuel Fort for their enthusiastic support during the development of the FCD method in their groups. I would also like to thank Guillaume Du Moulinet d’Hardemare and Lucie Domino for testing the method in their experiments and giving the necessary practical feedback.

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Appendix: Comparison to PIVlab

Comparison between displacement fields obtained by a PIVlab, b DIC+OF and c FCD operating on the soap bubble data. Only the horizontal displacement component is shown here. In d, horizontal traces through the center of the bubble, as indicated by the dashed line in c, are plotted. To facilitate comparison, the curves in d were slightly offset from each other

To perform DIC analysis in synthetic Schlieren applications, often out-of-the-box PIV software, developed for the tracking seeding particles in a flow, is employed. However, the imaging of optical distortions presents somewhat different challenges than the tracking of particles in a flow. It can therefore be expected that the performance of PIV optimized software is not necessarily optimal for SSI.

In PIV systems, for example, particle seeding conditions are often far from optimal, favoring robustness of the correlation algorithm over accuracy. In a typical SSI setup, on the other hand, one can simply create the perfect particle pattern, so that any loss in accuracy becomes directly apparent. Furthermore, particle displacements caused by a flow are often closer to the basic translation-only assumption underlying DIC, so that window deformations can be incorporated as a second-order effect (also the particles themselves do not deform in PIV). In contrast, in Schlieren imaging, the virtual strains (at the scale of the texture) are often very large.

To illustrate the above points, and to justify the use of a custom-made DIC+OF scheme in the comparison in the main text, a short comparison to the openly available PIVlab software (v. 1.43) (Thielicke and Stamhuis 2014) is included here. In Fig. 7, the horizontal distortions obtained by (a) PIVlab, (b) DIC+OF and (c) FCD for the popping bubble experiment at \(t = 74\,\) ms are shown. Figure 7d shows horizontal traces through the center of the bubble for each case. PIVlab and DIC+OF were fed images from the same experiment with a random dot background, while the FCD data was obtained in a separate experiment with a checkerboard background. In PIVlab, the three-stage multi-pass FFT cross correlation option was used, which applies a window deformation between each stage. The correlation window sizes for each stage were set to of \(16\times 16\), \(10\times 10\) and \(10\times 10\) pixels, respectively. PIVlab performs cross-correlation between equally sized window pairs (Thielicke and Stamhuis 2014), as opposed to the DIC+OF scheme, in which each sub-image in the reference is correlated with a larger search area in the distorted image.

Comparing Fig. 7a, b it is immediately clear that the DIC+OF method recovers the image displacements with significantly higher fidelity than PIVlab. Even after two window deformation cycles, PIVlab fails to resolve the strong optical distortions at the center of the image (showing NaN’s in that region). Furthermore, as is seen in Fig. 7d PIVlab overall reports somewhat smaller distortions than the DIC+OF and FCD schemes, and does not recover the smallest capillary waves. This is probably due to PIVlab’s bias to zero displacement, inherent to the correlation technique it employs (this underestimation effect can be reproduced by artificially shifting the reference by 1 pixel and then performing the analysis between the unshifted and shifted version, resulting in a reported mean displacement of 0.95 ± 0.06 px by PIVlab, while DIC+OF correctly returns 1.00 ± 0.04 px). Although PIVlab returned its result roughly five times faster than DIC+OF (although at a five times lower resolution), in terms of both speed and accuracy, the two correlation methods are still greatly outperformed by the FCD method (which was roughly 16x faster than PIVlab).